To begin, a simple check on the octal's group axioms - that they hold for the octal embedded in every "God-like" disjunction. As to follow, those many details are expounded upon as they appear!

The Group Axioms Fulfilled
There are a few results on the octal scattered throughout the book; here I present those results in this one page to verify that the group axioms hold for the octal in every disjunction.

Worlds Otherwise Empty
At the very start of my book "Seven Eyes Open" (4th Edition) I made some notes on Gödel's ontological argumnent; the most tantalising of which is the claim that the "empty essence lemma" applies across otherwise empty worlds every bit as much as does the disjunction solved with virtue in the octal. That the predicate of NE, or divine and necessary existence collapses to the Godlike essence alone is staggering, but more so is that the empty essence lemma does, in the case of an empty world, become confused with the empty middle of every disjunction made across such world-pairs where one pair is always an empty world. Such empty worlds do not "exist" but are then sophist in their possibility (The positive properties together form an ultrafilter). So, then, why the confusion over such an incongruence?

The Marriage Of The Lamb
In the book I simply state that the least's circuit and the consequent modal collapse is enough to justify the operation of the octal as a union of ultrafilters (I.e. all seven are equal yet are indexed differently). I take another (less brief) look at that here.

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